Subgroups — #149351

Order 512.

514 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20797	C₂	1	1	yes	maximal, characteristic
2	C₂×#1389	C₂	1	2	yes	maximal
3	C₂×#1384	C₂	1	2	yes	maximal
4	C₂×#1376	C₂	1	1	yes	maximal, characteristic
5	#10859	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₃₂	C₂²	1	2	yes	—
7	C₂²×#108	C₂²	1	1	yes	characteristic
8	C₂×#427	—	2	1	no	—
9	C₂×#427	C₂²	1	2	yes	—
10	C₂×#427	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	#2285	—	2	1	no	—
13	#1846	—	2	1	no	—
14	#1846	C₄	1	2	yes	complemented
15	#1845	—	2	1	no	—
16	#1844	C₄	1	2	yes	complemented
17	C₂×#339	—	2	1	no	—
18	#1729	—	2	1	no	—
19	#1389	—	2	4	no	—
20	#1384	—	2	4	no	—
21	#1376	—	2	1	no	—
22	#1376	—	2	1	no	—
23	C₂²×M₁₆	—	2	1	no	—
24	C₂³×C₈	D₄	1	1	yes	characteristic
25	C₂³×C₈	C₂³	1	1	yes	characteristic
26	C₂×#117	—	2	2	no	—
27	C₂×#117	—	2	1	no	—
28	C₂²×C₄²	D₄	1	1	yes	characteristic
29	C₂×M₃₂	—	2	2	no	—
30	C₂×M₃₂	—	2	2	no	—
31	C₂×M₃₂	—	2	2	no	—
32	C₂×M₃₂	—	2	2	no	—
33	C₂×M₃₂	—	2	2	no	—
34	C₂×M₃₂	D₄	1	4	yes	—
35	C₂²×C₁₆	—	2	2	no	—
36	C₂×#108	—	2	2	no	—
37	C₂×#108	—	2	2	no	—
38	C₂×#108	—	2	1	no	—
39	C₂×#108	Q₈	1	2	yes	—
40	#427	—	2	2	no	—
41	#427	—	2	2	no	—
42	#427	—	2	2	no	—
43	#427	—	2	2	no	—
44	#427	—	2	2	no	—
45	#427	—	2	2	no	—
46	#427	—	4	2	no	—
47	#427	—	4	2	no	—
48	C₂×#107	—	2	2	no	—
49	C₂×#107	—	2	2	no	—
50	C₂×#107	—	2	1	no	—
51	C₂×#106	—	2	2	no	—
52	C₂×#106	C₂×C₄	1	4	yes	—
53	C₂×#106	Q₈	1	2	yes	—
54	C₂×C₄×C₈	—	2	2	no	—
55	C₂×C₄×C₈	—	2	2	no	—
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	D₄	1	1	yes	characteristic
59	C₂×C₄×C₈	D₄	1	1	yes	characteristic
60	#384	—	2	2	no	—
61	#382	—	2	2	no	—
62	#380	—	4	2	no	—
63	C₄×C₄⋊C₄	—	4	2	no	—
64	#339	—	4	2	no	—
65	#336	—	4	2	no	—
66	C₂³×C₄	—	2	1	no	—
67	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂×M₁₆	—	2	2	no	—
69	C₂×M₁₆	—	4	1	no	—
70	C₂×M₁₆	—	4	1	no	—
71	C₂×M₁₆	—	4	1	no	—
72	C₂×M₁₆	—	4	1	no	—
73	C₂×M₁₆	—	4	1	no	—
74	C₂²×C₈	—	2	1	no	—
75	C₂²×C₈	—	2	1	no	—
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	—	2	1	no	—
78	C₂²×C₈	—	4	1	no	—
79	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₈	SD₁₆	1	2	yes	—
82	C₂²×C₈	Q₁₆	1	2	yes	—
83	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₈	C₂×D₄	1	2	yes	—
85	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₈	#41	1	2	yes	—
89	C₂²×C₈	#41	1	2	yes	—
90	C₂²×C₈	#41	1	2	yes	—
91	C₂²×C₈	#41	1	1	yes	characteristic
92	C₂²×C₈	#41	1	1	yes	characteristic
93	#117	—	4	1	no	—
94	#117	—	4	2	no	—
95	#117	—	4	1	no	—
96	#117	—	4	2	no	—
97	#117	—	4	1	no	—
98	#117	—	4	1	no	—
99	#117	—	4	2	no	—
100	#117	—	4	2	no	—
101	C₂×C₄⋊C₄	—	2	4	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	—	4	4	no	—
104	C₂×C₂²⋊C₄	—	4	1	no	—
105	C₂×C₂²⋊C₄	—	4	2	no	—
106	C₂×C₄²	—	2	1	no	—
107	C₂×C₄²	—	2	1	no	—
108	C₂×C₄²	—	2	1	no	—
109	C₂×C₄²	—	4	1	no	—
110	C₂×C₄²	—	4	2	no	—
111	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄²	SD₁₆	1	2	yes	—
114	C₂×C₄²	Q₁₆	1	2	yes	—
115	M₃₂	—	2	4	no	—
116	M₃₂	—	2	4	no	—
117	M₃₂	—	4	2	no	—
118	M₃₂	—	4	2	no	—
119	C₂×C₁₆	—	2	4	no	—
120	C₂×C₁₆	—	4	2	no	—
121	C₂×C₁₆	—	4	2	no	—
122	#108	—	2	4	no	—
123	#108	—	4	2	no	—
124	#107	—	4	4	no	—
125	#107	—	4	2	no	—
126	#107	—	4	2	no	—
127	#107	—	4	2	no	—
128	#106	—	2	4	no	—
129	#106	—	2	4	no	—
130	#106	—	2	4	no	—
131	#106	—	4	4	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	2	2	no	—
136	#95	—	4	4	no	—
137	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	—	4	4	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
147	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
148	C₂²×C₄	#123	1	1	yes	characteristic
149	C₂²×C₄	#123	1	1	yes	characteristic
150	C₂²×C₄	#124	1	1	yes	characteristic
151	C₂²×C₄	#127	1	1	yes	characteristic
152	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
153	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
154	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
155	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
156	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
159	M₁₆	—	4	2	no	—
160	M₁₆	—	4	2	no	—
161	M₁₆	—	8	1	no	—
162	M₁₆	—	8	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	8	1	no	—
173	C₂×C₈	—	8	1	no	—
174	C₂×C₈	#103	1	2	yes	—
175	C₂×C₈	#103	1	2	yes	—
176	C₂×C₈	#117	1	2	yes	—
177	C₂×C₈	C₄×D₄	1	2	yes	—
178	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
179	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
180	C₂×C₈	#121	1	2	yes	—
181	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
182	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
183	C₂×C₈	#123	1	2	yes	—
184	C₂×C₈	#123	1	2	yes	—
185	C₂×C₈	#123	1	2	yes	—
186	C₂×C₈	#124	1	1	yes	characteristic
187	C₂×C₈	#125	1	2	yes	—
188	C₂×C₈	#125	1	2	yes	—
189	C₂×C₈	#126	1	2	yes	—
190	C₂×C₈	#126	1	2	yes	—
191	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
192	C₄⋊C₄	—	4	4	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	8	8	no	—
197	C₂²⋊C₄	—	8	1	no	—
198	C₂²⋊C₄	—	8	2	no	—
199	C₂²⋊C₄	—	8	1	no	—
200	C₂²⋊C₄	—	8	2	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	8	1	no	—
206	C₄²	—	8	2	no	—
207	C₄²	—	8	1	no	—
208	C₄²	—	8	2	no	—
209	C₄²	#103	1	2	yes	—
210	C₄²	#103	1	2	yes	—
211	C₁₆	—	4	4	no	—
212	C₂³	—	4	2	no	—
213	C₂³	#418	1	1	yes	characteristic
214	C₂³	#476	1	1	yes	characteristic
215	C₂³	#477	1	1	yes	characteristic
216	C₂³	#481	1	1	yes	characteristic
217	C₂³	#484	1	1	yes	characteristic
218	C₂³	#490	1	1	yes	characteristic
219	C₂³	#496	1	1	yes	characteristic
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	8	8	no	—
227	C₂×C₄	—	8	4	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	4	no	—
230	C₂×C₄	#381	1	1	yes	characteristic
231	C₂×C₄	#390	1	1	yes	characteristic
232	C₂×C₄	#391	1	1	yes	characteristic
233	C₂×C₄	#399	1	1	yes	characteristic
234	C₂×C₄	#399	1	1	yes	characteristic
235	C₂×C₄	#400	1	1	yes	characteristic
236	C₂×C₄	#400	1	1	yes	characteristic
237	C₂×C₄	C₂×#103	1	1	yes	characteristic
238	C₂×C₄	C₂×#103	1	1	yes	characteristic
239	C₂×C₄	#418	1	1	yes	characteristic
240	C₂×C₄	#476	1	1	yes	characteristic
241	C₂×C₄	#476	1	1	yes	characteristic
242	C₂×C₄	#476	1	1	yes	characteristic
243	C₂×C₄	#477	1	1	yes	characteristic
244	C₂×C₄	#477	1	1	yes	characteristic
245	C₂×C₄	#477	1	1	yes	characteristic
246	C₂×C₄	#481	1	1	yes	characteristic
247	C₂×C₄	#481	1	1	yes	characteristic
248	C₂×C₄	#481	1	1	yes	characteristic
249	C₂×C₄	#484	1	1	yes	characteristic
250	C₂×C₄	#484	1	1	yes	characteristic
251	C₂×C₄	#484	1	1	yes	characteristic
252	C₂×C₄	#487	1	1	yes	characteristic
253	C₂×C₄	#487	1	1	yes	characteristic
254	C₂×C₄	#490	1	1	yes	characteristic
255	C₂×C₄	#492	1	1	yes	characteristic
256	C₂×C₄	#494	1	1	yes	characteristic
257	C₂×C₄	#496	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₈	—	8	2	no	—
261	C₈	#387	1	2	yes	—
262	C₈	#387	1	2	yes	—
263	C₈	#397	1	2	yes	—
264	C₈	#398	1	2	yes	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	1	no	—
272	C₂²	#1824	1	1	yes	characteristic
273	C₂²	#1961	1	1	yes	characteristic
274	C₂²	#1987	1	1	yes	characteristic
275	C₂²	#2094	1	1	yes	characteristic
276	C₂²	#2095	1	1	yes	characteristic
277	C₂²	#2100	1	1	yes	characteristic
278	C₂²	#2100	1	1	yes	characteristic
279	C₄	—	2	4	no	—
280	C₄	—	4	2	no	—
281	C₄	—	16	4	no	—
282	C₄	—	16	2	no	—
283	C₄	#1445	1	1	yes	characteristic
284	C₄	#1824	1	1	yes	characteristic
285	C₄	#1956	1	1	yes	characteristic
286	C₄	#1987	1	1	yes	characteristic
287	C₄	#2094	1	1	yes	characteristic
288	C₄	#2095	1	1	yes	characteristic
289	C₄	#2100	1	1	yes	characteristic
290	C₄	#2100	1	1	yes	characteristic
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	2	1	no	—
297	C₂	#11371	1	1	yes	characteristic
298	C₂	#13271	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1987
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
14. #1846	C₄	semidirect
16. #1844	C₄	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20797	2
C₂×#1389	2
C₂×#1384	2
C₂×#1376	2
#10859	2